Stability And Switching-based Stabilization For Circulant Systems

Circulant systems are a class of systems possessing great significance in practice. The circulant structure of this class of systems determines its many characteristics. Circulant systems need to be given special analysis and design. The study of circulant systems not only shows the impact of the circulant structure on system's dynamic behavior and thus the circulant structure can be effectively used to derive useful properties of systems, but also produces certain effects for analysis and design of composite systems and large-scale systems. Therefore, the study of circulant systems has drawn particular attention.This dissertation studies the circulant systems which are extended from the models of the hypercycles. Stability and switching-based stabilization are particularly studied using Lyapunov' stability theory. The main content of the paper is summarized as follows.The modeling of circulant systems from the hypercycles in biology is explained in the first chapter. The research developments of circulant systems and switched systems are reviewed.In the second chapter, the structural properties of circulant matrixes are discussed and some foundational knowledge is reviewed, which are required for research in the dissertation. Special emphasis is put on the property that a circulant matrix is similar to a diagonal matrix by an invertible matrix, which is independent of circulant elements, and that the Lyapunov' equation associated circulant matrix has a solution, which is also independent of circulant elements. Then, circulant systems are described by state equations.In the third chapter, by means of the structural properties of circulant matrixes, a state transformation is found, which makes states of linear circulant systems decoupled. Thus, sufficient and necessary conditions of asymptotic stability of the linear circulant systems, including time varying and time invariant cases, are obtained. The robust stable region for the linear uncertain circulant systems is given.